### Summary

Bernoulli's principle relates the pressure of a fluid to its elevation and its speed. Bernoulli's equation can be used to approximate these parameters in water, air or any fluid that has very low viscosity. Students learn about the relationships between the components of the Bernoulli equation through real-life engineering examples and practice problems.

### Engineering Connection

### Educational Standards

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### Pre-Req Knowledge

### Learning Objectives

- Calculate an unknown fluid condition (for example, fluid pressure, velocity, density or height) at one point along a flow streamline, if conditions are known at another point along the same streamline.
- Use the Bernoulli equation to explain that faster airflow causes a decrease in pressure, and give an example of a real-life application.

### Introduction/Motivation

^{}

*v*is fluid velocity,

*ρ*is fluid density,

*z*is relative height, and

*P*is pressure. Applying this equation to an example helps to make it clearer. Consider a reservoir located up in the mountains with a pipe leading down to a town at a lower elevation. The pipe delivers water to a hydroelectric plant, and we want to know how fast the water will flow into the plant turbines. Figure 2 illustrates this situation.

### Lesson Background and Concepts for Teachers

*v*is fluid velocity,

*ρ*is fluid density,

*h*is relative height, and

*P*is pressure. Notice the constant has units of pressure as well.

*KE*is kinetic energy,

*PE*is potential energy, and

*W*is the work done on the system. Imagine a block of ice sliding down the water slide at some velocity. The block has a kinetic energy equal to one-half its mass times its velocity squared, or

*KE*is kinetic energy and

*m*is mass. The block also has a certain potential energy described by

*PE*is the potential energy and

*h*is the height. Finally, the work is being done on the block by the force of the water pressure behind it (with a force

*F=PA*, where

*P*is water pressure, and

*A*is the area of the face of the ice that is getting pushed along). When the block is pushed a distance equivalent to its own width,

*Δx*, then the work done on the block is

*V*is volume. The equation for conservation of energy becomes

*ρ*Δ equals mass divided by volume, it reduces to

### Vocabulary/Definitions

Bernoulli Principle: |
In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. Named after Dutch-Swiss mathematician Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738. Also called the Bernoulli effect. |

inviscid flow: |
Flow in which one can ignore the effects of fluid viscosity. |

streamline: |
A line tangent to the flow of a fluid at any given instant. |

Venturi effect: |
The reduction in fluid pressure that results when a fluid flows through a constricted section of pipe. As a fluid's velocity increases, its pressure decreases, and vice versa. Named after Italian physicist Giovanni Battista Venturi (1746–1822). |

### Associated Activities

- A Shot Under Pressure - Students use their understanding of projectile physics and fluid dynamics to find the water pressure in a water gun. By measuring the range of the water jet, they are able to calculate the theoretical pressure. Students create graphs to analyze how the predicted pressure relates to the number of times they pump the powerful squirt gun.

### Lesson Closure

### Attachments

- Bernoulli Equation Practice Worksheet (doc)
- Bernoulli Equation Practice Worksheet (pdf)
- Bernoulli Equation Practice Worksheet Answers (doc)
- Bernoulli Equation Practice Worksheet Answers (pdf)
- Bernoulli Flow Graphics (suitable for overhead transparencies or handouts) (ppt)
- Bernoulli Flow Graphics (suitable for overhead transparencies or handouts) (pdf)

### Assessment

Pre-Lesson Assessment

*Matching:*To help students with this lesson and its associated activity, create a list of physics equations that they have already studied, such as kinetic energy, potential energy, work, kinematics equations. Randomly write the physics terms on the left side of the board and the matching halves of the equations (out of order) on the right side of the board. As a class, have students match the correct sides together. Examples:

Post-Introduction Assessment

*Homework/Independent Practice:*Have students complete the Bernoulli Equation Practice Worksheet. Have all students complete the first problem in class and review the answer together. Have students complete the second problem as homework.

Lesson Summary Assessment

*Discussion Question:*During the next class period, lead a five-minute discussion asking students what they learned from the homework assignment.

### Additional Multimedia Support

### References

Bernoulli's principle (definition). Last updated February 11, 2010. Wikipedia, The Free Encyclopedia. Accessed February 17, 2010. http://en.wikipedia.org/w/index.php?title=Bernoulli%27s_principle&oldid=343435891

Knight, Randall. *Physics for Scientists and Engineers: a Strategic Approach*. Second edition. San Francisco, CA: Pearson Addison-Wesley, 2008.

Munson, B. R., Young, D.F., Okiishi, T.H. *Fundamentals of Fluid Mechanics*. Fifth edition. New York, NY: John Wiley & Sons, Inc., 2006.

Venturi effect (definition). Last updated February 12, 2010. Wikipedia, The Free Encyclopedia. Accessed February 17, 2010. http://en.wikipedia.org/w/index.php?title=Venturi_effect&oldid=343508711

### Contributors

James Prager, Karen King, Denise W. Carlson

### Copyright

© 2009 by Regents of the University of Colorado.

### Supporting Program

Integrated Teaching and Learning Program and Laboratory, College of Engineering, University of Colorado Boulder

### Acknowledgements

Last modified: June 30, 2015